Finite depth and Jacobson-Bourbaki correspondence

We introduce a notion of depth three tower C⊆B⊆A with depth two ring extension A|B being the case B=C. If A=EndBC and B|C is a Frobenius extension with A|B|C depth three, then A|C is depth two. If A, B and C correspond to a tower G>H>K via group algebras over a base ring F, the depth three condition is the condition that K has normal closure KG contained in H. For a depth three tower of rings, a pre-Galois theory for the ring EndACB and coring (A⊗BA)C involving Morita context bimodules and left coideal subrings is applied to specialize a Jacobson-Bourbaki correspondence theorem for augmented rings to depth two extensions with depth three intermediate division rings.